This establishes \(L\)  in (REF ) as a valid loss for an order-coherent belief update. However, \(\pi _0\)  is fixed in the variation over \(\nu \)  in (REF ), so (REF ) is the density of the measure \(\nu \)  maximising (REF ) for both additive and prequentially additive losses. The proof of this step is unchanged from that given at the end of Section 1.1 of [1]}.
